If you haven't learnt about Bezier curves already, go ahead and do so now.

Add a bezier curve, and tab into edit mode. Now you have to model the path by extruding and subdividing the curve. The shape of the curve defines the path of the particles. After you are done with the curve, you can proceed to create the emission object.

Add a UV Sphere with 32 segments and 16 rings. This is going to be the emitter, i.e. the object that gives out those coloured spheres.

This is going to have a completely transparent material since hiding it hides the whole thing.

So go to material properties, add a new material. Scroll down to the Transparency panel, check it and change the alpha value to zero, making it completely transparent. Also change the specular intensity to zero. Leave everything else as such.

In this part, you are going to make the sphere follow the curve you had modeled earlier. Select the cube, and add a Follow Path Constraint. Select the target, i.e. the curve and click on Animate Path. All these steps speak for themselves, there's no need for me to explain. You could check follow curve, but since this is a sphere this is not needed. All it does is make the object face the direction it is moving towards.

As for the speed of the object, you can change it by -

Select the curve.

Go to Object properties -

Scroll down to Path Animation panel.

Change Path Animation Frames value to the no. of frames you want the animation to last. I prefer it to be 150.

In the end, make a few changes until you are satisfied with the result. Now you are ready to proceed to part 2.